In this paper, we study the optimal cooperative beamforming problem in the amplify-and-forward (AF) multi-user and multi-relay networks, which face the challenges of rapid node number variations and per-node power limits.To achieve extra diversity gain, direct-link (source-to-destination) and distributed relay-link signals are jointly exploited.The optimal cooperative beamforming problem is 30x24x24 wall cabinet formulated as the maximum relay transmit power minimization problem, subject to per-relay transmit power and the minimum destination signal-to-noise ratio (SNR) constraints.Since the problem is non-convex, we introduce a phase-regulation (PR) method to transform the non-convex problem into a tractable second-order cone programming (SOCP) problem.It is demonstrated that the proposed method can provide much more robustness against node number variations in terms of worst-case convergence rates than the Lagrange dual and teal horse blanket the successive convex approximation (SCA) methods.
Furthermore, the closed-form expressions of two necessary feasibility conditions are derived, by which the infeasible channels can be identified and excluded.Consequently, computational costs are reduced.The equivalence of the proposed Necessary Condition I (NC1) and the signal-to-interference ratio (SIR) condition is proved theoretically and numerically.The proposed Necessary Condition II (NC2) has a lower upper-bound than the SIR condition, thus reducing more computational costs.This method is applicable to both direct-link and non-direct-link scenarios.